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Commutators of small rank and reducibility of operator semigroups


Authors: Ali Jafarian, Alexey I. Popov, Mehdi Radjabalipour and Heydar Radjavi
Journal: Proc. Amer. Math. Soc. 142 (2014), 4277-4289
MSC (2010): Primary 47D03, 20M20; Secondary 47B47, 51F25
DOI: https://doi.org/10.1090/S0002-9939-2014-12217-9
Published electronically: August 13, 2014
MathSciNet review: 3266995
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Abstract: It is easy to see that if $ \mathcal {G}$ is a non-abelian group of unitary matrices, then for no members $ A$ and $ B$ of $ \mathcal {G}$ can the rank of $ AB-BA$ be one. We examine the consequences of the assumption that this rank is at most two for a general semigroup $ \mathcal {S}$ of linear operators. Our conclusion is that under obviously necessary, but trivial, size conditions, $ \mathcal {S}$ is reducible. In the case of a unitary group satisfying the hypothesis, we show that it is contained in the direct sum $ \mathcal {G}_1\oplus \mathcal {G}_2$, where $ \mathcal {G}_1$ is at most $ 3\times 3$ and $ \mathcal {G}_2$ is abelian.


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  • [1] Janez Bernik, Robert Guralnick, and Mitja Mastnak, Reduction theorems for groups of matrices, Linear Algebra Appl. 383 (2004), 119-126. MR 2073898 (2005f:20080), https://doi.org/10.1016/j.laa.2003.11.020
  • [2] Grega Cigler, Roman Drnovšek, Damjana Kokol-Bukovšek, Matjaž Omladič, Thomas J. Laffey, Heydar Radjavi, and Peter Rosenthal, Invariant subspaces for semigroups of algebraic operators, J. Funct. Anal. 160 (1998), no. 2, 452-465. MR 1665294 (2000b:47015), https://doi.org/10.1006/jfan.1998.3293
  • [3] Roman Drnovšek, Hyperinvariant subspaces for operator semigroups with commutators of rank at most one, Houston J. Math. 26 (2000), no. 3, 543-548. MR 1811940 (2002d:47007)
  • [4] Roman Drnovšek, Invariant subspaces for operator semigroups with commutators of rank at most one, J. Funct. Anal. 256 (2009), no. 12, 4187-4196. MR 2521924 (2010e:47019), https://doi.org/10.1016/j.jfa.2009.03.010
  • [5] Mitja Mastnak and Heydar Radjavi, Structure of finite, minimal nonabelian groups and triangularization, Linear Algebra Appl. 430 (2009), no. 7, 1838-1848. MR 2494668 (2010c:20005), https://doi.org/10.1016/j.laa.2008.09.018
  • [6] Heydar Radjavi and Peter Rosenthal, From local to global triangularization, J. Funct. Anal. 147 (1997), no. 2, 443-456. MR 1454489 (98j:47010), https://doi.org/10.1006/jfan.1996.3069
  • [7] Heydar Radjavi and Peter Rosenthal, Simultaneous triangularization, Universitext, Springer-Verlag, New York, 2000. MR 1736065 (2001e:47001)

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Additional Information

Ali Jafarian
Affiliation: University of New Haven, 300 Boston Post Road, West Haven, Connecticut 06516
Email: ajafarian@newhaven.edu

Alexey I. Popov
Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L3G1, Canada
Email: a4popov@uwaterloo.ca

Mehdi Radjabalipour
Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L3G1, Canada (on sabbatical from the Iranian Academy of Sciences, Tehran, Iran)
Email: radjabalipour@ias.ac.ir

Heydar Radjavi
Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L3G1, Canada
Email: hradjavi@uwaterloo.ca

DOI: https://doi.org/10.1090/S0002-9939-2014-12217-9
Keywords: Semigroup of operators, unitary group, commutator, rank, invariant subspace
Received by editor(s): January 16, 2013
Published electronically: August 13, 2014
Additional Notes: The second and fourth authors’ research was supported in part by NSERC (Canada)
The third author’s research was supported in part by the Iranian National Science Foundation
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2014 American Mathematical Society

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